Topological group

A homomorphism of topological groups means a continuous group homomorphism G → H. Topological groups, together with their homomorphisms, form a category. A group homomorphism between topological groups is continuous if and only if it is continuous at some point.[4]

An isomorphism of topological groups is a group isomorphism that is also a homeomorphism of the underlying topological spaces. This is stronger than simply requiring a continuous group isomorphism—the inverse must also be continuous. There are examples of topological groups that are isomorphic as ordinary groups but not as topological groups. Indeed, any non-discrete topological group is also a topological group when considered with the discrete topology. The underlying groups are the same, but as topological groups there is not an isomorphism.

Every topological group's topology is translation invariant, which by definition means that if for any ${\displaystyle a\in G,}$ left or right multiplication by this element yields a homeomorphism ${\displaystyle G\to G.}$ Consequently, for any ${\displaystyle a\in G}$ and ${\displaystyle S\subseteq G,}$ the subset ${\displaystyle S}$ is open (resp. closed) in ${\displaystyle G}$ if and only if this is true of its left translation ${\displaystyle aS:=\{as:s\in S\}}$ and right translation ${\displaystyle Sa:=\{sa:s\in S\}.}$ If ${\displaystyle {\mathcal {N}}}$ is a neighborhood basis of the identity element in a topological group ${\displaystyle G}$ then for all ${\displaystyle x\in X,}$ ${\displaystyle x{\mathcal {N}}:=\{xN:N\in {\mathcal {N}}\}}$ is a neighborhood basis of ${\displaystyle x}$ in ${\displaystyle G.}$[4] In particular, any group topology on a topological group is completely determined by any neighborhood basis at the identity element. If ${\displaystyle S}$ is any subset of ${\displaystyle G}$ and ${\displaystyle U}$ is an open subset of ${\displaystyle G,}$ then ${\displaystyle SU:=\{su:s\in S,u\in U\}}$ is an open subset of ${\displaystyle G.}$[4]

The inversion operation ${\displaystyle g\mapsto g^{-1}}$ on a topological group ${\displaystyle G}$ is a homeomorphism from ${\displaystyle G}$ to itself.

A subset ${\displaystyle S\subseteq G}$ is said to be symmetric if ${\displaystyle S^{-1}=S,}$ where ${\displaystyle S^{-1}:=\left\{s^{-1}:s\in S\right\}.}$ The closure of every symmetric set in a commutative topological group is symmetric.[4] If S is any subset of a commutative topological group G, then the following sets are also symmetric: S⁻¹ ∩ S, S⁻¹ ∪ S, and S⁻¹ S.[4]

For any neighborhood N in a commutative topological group G of the identity element, there exists a symmetric neighborhood M of the identity element such that M⁻¹ M ⊆ N, where note that M⁻¹ M is necessarily a symmetric neighborhood of the identity element.[4] Thus every topological group has a neighborhood basis at the identity element consisting of symmetric sets.

If G is a locally compact commutative group, then for any neighborhood N in G of the identity element, there exists a symmetric relatively compact neighborhood M of the identity element such that cl M ⊆ N (where cl M is symmetric as well).[4]

Every topological group can be viewed as a uniform space in two ways; the left uniformity turns all left multiplications into uniformly continuous maps while the right uniformity turns all right multiplications into uniformly continuous maps.[5] If G is not abelian, then these two need not coincide. The uniform structures allow one to talk about notions such as completeness, uniform continuity and uniform convergence on topological groups.

If U is an open subset of a commutative topological group G and U contains a compact set K, then there exists a neighborhood N of the identity element such that KN ⊆ U.[4]

As a uniform space, every commutative topological group is completely regular. Consequently, for a multiplicative topological group G with identity element 1, the following are equivalent:[4]

1. G is a T₀-space (Kolmogorov);
2. G is a T₂-space (Hausdorff);
3. G is a T_31⁄2 (Tychonoff);
4. { 1 } is closed in G;
5. { 1 } := ∩N ∈ 𝒩 N, where 𝒩 is a neighborhood basis of the identity element in G;
6. for any ${\displaystyle x\in G}$ such that ${\displaystyle x\neq 1,}$ there exists a neighborhood U in G of the identity element such that ${\displaystyle x\not \in U.}$

A subgroup of a commutative topological group is discrete if and only if it has an isolated point.[4]

If G is not Hausdorff, then one can obtain a Hausdorff group by passing to the quotient group G/K, where K is the closure of the identity.[6] This is equivalent to taking the Kolmogorov quotient of G.

Let G be a topological group. As with any topological space, we say that G is metrisable if and only if there exists a metric d on G, which induces the same topology on ${\displaystyle G}$. A metric d on G is called

• left-invariant (resp. right-invariant) if and only if ${\displaystyle d(ax_{1},ax_{2})=d(x_{1},x_{2})}$(resp. ${\displaystyle d(x_{1}a,x_{2}a)=d(x_{1},x_{2})}$) for all ${\displaystyle a,x_{1},x_{2}\in G}$ (equivalently, ${\displaystyle d}$ is left-invariant just in case the map ${\displaystyle x\mapsto ax}$ is an isometry from ${\displaystyle (G,d)}$ to itself for each ${\displaystyle a\in G}$).
• proper if and only if all open balls, ${\displaystyle B(r)=\{g\in G\mid d(g,1)<r\}}$ for ${\displaystyle r>0}$, are pre-compact.

The Birkhoff–Kakutani theorem (named after mathematicians Garrett Birkhoff and Shizuo Kakutani) states that the following three conditions on a topological group G are equivalent:[7]

1. G is first countable (equivalently: the identity element 1 is closed in G, and there is a countable basis of neighborhoods for 1 in G).
2. G is metrisable (as a topological space).
3. There is a left-invariant metric on G that induces the given topology on G.

Furthermore, the following are equivalent for any topological group G:

1. G is a second countable locally compact (Hausdorff) space.
2. G is a Polish, locally compact (Hausdorff) space.
3. G is properly metrisable (as a topological space).
4. There is a left-invariant, proper metric on G that induces the given topology on G.

Note: As with the rest of the article we of assume here a Hausdorff topology. The implications 4 ${\displaystyle \Rightarrow }$ 3 ${\displaystyle \Rightarrow }$ 2 ${\displaystyle \Rightarrow }$ 1 hold in any topological space. In particular 3 ${\displaystyle \Rightarrow }$ 2 holds, since in particular any properly metrisable space is countable union of compact metrisable and thus separable (cf. properties of compact metric spaces) subsets. The non-trivial implication 1 ${\displaystyle \Rightarrow }$ 4 was first proved by Raimond Struble in 1974.[8] An alternative approach was made by Uffe Haagerup and Agata Przybyszewska in 2006,[9] the idea of the which is as follows: One relies on the construction of a left-invariant metric, ${\displaystyle d_{0}}$, as in the case of first countable spaces. By local compactness, closed balls of sufficiently small radii are compact, and by normalising we can assume this holds for radius 1. Closing the open ball, U, of radius 1 under multiplication yields a clopen subgroup, H, of G, on which the metric ${\displaystyle d_{0}}$ is proper. Since H is open and G is second countable, the subgroup has at most countably many cosets. One now uses this sequence of cosets and the metric on H to construct a proper metric on G.

Every subgroup of a topological group is itself a topological group when given the subspace topology. Every open subgroup H is also closed in G, since the complement of H is the open set given by the union of open sets gH for g ∈ G \ H. If H is a subgroup of G then the closure of H is also a subgroup. Likewise, if H is a normal subgroup of G, the closure of H is normal in G.

If H is a subgroup of G, the set of left cosets G/H with the quotient topology is called a homogeneous space for G. The quotient map ${\displaystyle q:G\to G/H}$ is always open. For example, for a positive integer n, the sphere Sⁿ is a homogeneous space for the rotation group SO(n+1) in ${\displaystyle \mathbb {R} }$ⁿ⁺¹, with Sⁿ = SO(n+1)/SO(n). A homogeneous space G/H is Hausdorff if and only if H is closed in G.[10] Partly for this reason, it is natural to concentrate on closed subgroups when studying topological groups.

If H is a normal subgroup of G, then the quotient group G/H becomes a topological group when given the quotient topology. It is Hausdorff if and only if H is closed in G. For example, the quotient group ${\displaystyle \mathbb {R} /\mathbb {Z} }$ is isomorphic to the circle group S¹.

In any topological group, the identity component (i.e., the connected component containing the identity element) is a closed normal subgroup. If C is the identity component and a is any point of G, then the left coset aC is the component of G containing a. So the collection of all left cosets (or right cosets) of C in G is equal to the collection of all components of G. It follows that the quotient group G/C is totally disconnected.[11]

In any commutative topological group, the product (assuming the group is multiplicative) KC of a compact set K and a closed set C is a closed set.[4] Furthermore, for any subsets R and S of G, (cl R)(cl S) ⊆ cl (RS).[4]

If H is a subgroup of a commutative topological group G and if N is a neighborhood in G of the identity element such that H ∩ cl N is closed, then H is closed.[4] Every discrete subgroup of a Hausdorff commutative topological group is closed.[4]

The isomorphism theorems from ordinary group theory are not always true in the topological setting. This is because a bijective homomorphism need not be an isomorphism of topological groups.

For example, a native version of the first isomorphism theorem is false for topological groups: if ${\displaystyle f:G\to H}$ is a morphism of topological groups (that is, a continuous homomorphism), it is not necessarily true that the induced homomorphism ${\displaystyle {\tilde {f}}:G/\ker f\to \mathrm {Im} (f)}$ is an isomorphism of topological groups; it will be a bijective, continuous homomorphism, but it will not necessarily be a homeomorphism. In other words, it will not necessarily admit an inverse in the category of topological groups.

There is a version of the first isomorphism theorem for topological groups, which may be stated as follows: if ${\displaystyle f:G\to H}$ is a continuous homomorphism, then the induced homomorphism from G/ker(f) to im(f) is an isomorphism if and only if the map f is open onto its image.[12]

The third isomorphism theorem, however, is true more or less verbatim for topological groups, as one may easily check.

This article will henceforth assume that any topological group that we consider is an additive commutative topological group with identity element ${\displaystyle 0.}$

The diagonal of ${\displaystyle X}$ is the set

$${\displaystyle \Delta _{X}:=\{(x,x):x\in X\}}$$

and for any ${\displaystyle N\subseteq X}$ containing ${\displaystyle 0,}$ the canonical entourage or canonical vicinities around ${\displaystyle N}$ is the set

$${\displaystyle \Delta _{X}(N):=\{(x,y)\in X\times X:x-y\in N\}=\bigcup _{y\in X}[(y+N)\times \{y\}]=\Delta _{X}+(N\times \{0\})}$$

For a topological group ${\displaystyle (X,\tau ),}$ the canonical uniformity[21] on ${\displaystyle X}$ is the uniform structure induced by the set of all canonical entourages ${\displaystyle \Delta (N)}$ as ${\displaystyle N}$ ranges over all neighborhoods of ${\displaystyle 0}$ in ${\displaystyle X.}$

That is, it is the upward closure of the following prefilter on ${\displaystyle X\times X,}$

$${\displaystyle \left\{\Delta (N):N{\text{ is a neighborhood of }}0{\text{ in }}X\right\}}$$

where this prefilter forms what is known as a base of entourages of the canonical uniformity.

For a commutative additive group ${\displaystyle X,}$ a fundamental system of entourages ${\displaystyle {\mathcal {B}}}$ is called a translation-invariant uniformity if for every ${\displaystyle B\in {\mathcal {B}},}$ ${\displaystyle (x,y)\in B}$ if and only if ${\displaystyle (x+z,y+z)\in B}$ for all ${\displaystyle x,y,z\in X.}$ A uniformity ${\displaystyle {\mathcal {B}}}$ is called translation-invariant if it has a base of entourages that is translation-invariant.[22]

• The canonical uniformity on any commutative topological group is translation-invariant.
• The same canonical uniformity would result by using a neighborhood basis of the origin rather the filter of all neighborhoods of the origin.
• Every entourage ${\displaystyle \Delta _{X}(N)}$ contains the diagonal ${\displaystyle \Delta _{X}:=\Delta _{X}(\{0\})=\{(x,x):x\in X\}}$ because ${\displaystyle 0\in N.}$
• If ${\displaystyle N}$ is symmetric (that is, ${\displaystyle -N=N}$) then ${\displaystyle \Delta _{X}(N)}$ is symmetric (meaning that ${\displaystyle \Delta _{X}(N)^{\operatorname {op} }=\Delta _{X}(N)}$) and ${\displaystyle \Delta _{X}(N)\circ \Delta _{X}(N)=\{(x,z):{\text{ there exists }}y\in X{\text{ such that }}x,z\in y+N\}=\bigcup _{y\in X}[(y+N)\times (y+N)]=\Delta _{X}+(N\times N).}$
• The topology induced on ${\displaystyle X}$ by the canonical uniformity is the same as the topology that ${\displaystyle X}$ started with (that is, it is ${\displaystyle \tau }$).

The general theory of uniform spaces has its own definition of a "Cauchy prefilter" and "Cauchy net." For the canonical uniformity on ${\displaystyle X,}$ these reduces down to the definition described below.

Suppose ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ is a net in ${\displaystyle X}$ and ${\displaystyle y_{\bullet }=\left(y_{j}\right)_{j\in J}}$ is a net in ${\displaystyle Y.}$ Make ${\displaystyle I\times J}$ into a directed set by declaring ${\displaystyle (i,j)\leq \left(i_{2},j_{2}\right)}$ if and only if ${\displaystyle i\leq i_{2}{\text{ and }}j\leq j_{2}.}$ Then[23] ${\displaystyle x_{\bullet }\times y_{\bullet }:=\left(x_{i},y_{j}\right)_{(i,j)\in I\times J}}$ denotes the product net. If ${\displaystyle X=Y}$ then the image of this net under the addition map ${\displaystyle X\times X\to X}$ denotes the sum of these two nets:

$${\displaystyle x_{\bullet }+y_{\bullet }:=\left(x_{i}+y_{j}\right)_{(i,j)\in I\times J}}$$

and similarly their difference is defined to be the image of the product net under the subtraction map:

$${\displaystyle x_{\bullet }-y_{\bullet }:=\left(x_{i}-y_{j}\right)_{(i,j)\in I\times J}.}$$

A net ${\displaystyle x_{\bullet }=\left(x_{i}\right)_{i\in I}}$ in an additive topological group ${\displaystyle X}$ is called a Cauchy net if[24]

$${\displaystyle \left(x_{i}-x_{j}\right)_{(i,j)\in I\times I}\to 0{\text{ in }}X}$$

or equivalently, if for every neighborhood ${\displaystyle N}$ of ${\displaystyle 0}$ in ${\displaystyle X,}$ there exists some ${\displaystyle i_{0}\in I}$ such that ${\displaystyle x_{i}-x_{j}\in N}$ for all indices ${\displaystyle i,j\geq i_{0}.}$

A Cauchy sequence is a Cauchy net that is a sequence.

If ${\displaystyle B}$ is a subset of an additive group ${\displaystyle X}$ and ${\displaystyle N}$ is a set containing ${\displaystyle 0,}$ then${\displaystyle B}$ is said to be an ${\displaystyle N}$-small set or small of order ${\displaystyle N}$ if ${\displaystyle B-B\subseteq N.}$[25]

A prefilter ${\displaystyle {\mathcal {B}}}$ on an additive topological group ${\displaystyle X}$ called a Cauchy prefilter if it satisfies any of the following equivalent conditions:

1. ${\displaystyle {\mathcal {B}}-{\mathcal {B}}\to 0}$ in ${\displaystyle X,}$ where ${\displaystyle {\mathcal {B}}-{\mathcal {B}}:=\{B-C:B,C\in {\mathcal {B}}\}}$ is a prefilter.
2. ${\displaystyle \{B-B:B\in {\mathcal {B}}\}\to 0}$ in ${\displaystyle X,}$ where ${\displaystyle \{B-B:B\in {\mathcal {B}}\}}$ is a prefilter equivalent to ${\displaystyle {\mathcal {B}}-{\mathcal {B}}.}$
3. For every neighborhood ${\displaystyle N}$ of ${\displaystyle 0}$ in ${\displaystyle X,}$ ${\displaystyle {\mathcal {B}}}$ contains some ${\displaystyle N}$-small set (that is, there exists some ${\displaystyle B\in {\mathcal {B}}}$ such that ${\displaystyle B-B\subseteq N}$).[25]

and if ${\displaystyle X}$ is commutative then also:

4. For every neighborhood ${\displaystyle N}$ of ${\displaystyle 0}$ in ${\displaystyle X,}$ there exists some ${\displaystyle B\in {\mathcal {B}}}$ and some ${\displaystyle x\in X}$ such that ${\displaystyle B\subseteq x+N.}$[25]

• It suffices to check any of the above condition for any given neighborhood basis of ${\displaystyle 0}$ in ${\displaystyle X.}$

Suppose ${\displaystyle {\mathcal {B}}}$ is a prefilter on a commutative topological group ${\displaystyle X}$ and ${\displaystyle x\in X.}$ Then ${\displaystyle {\mathcal {B}}\to x}$ in ${\displaystyle X}$ if and only if ${\displaystyle x\in \operatorname {cl} {\mathcal {B}}}$ and ${\displaystyle {\mathcal {B}}}$ is Cauchy.[23]

Recall that for any ${\displaystyle S\subseteq X,}$ a prefilter ${\displaystyle {\mathcal {C}}}$ on ${\displaystyle S}$ is necessarily a subset of ${\displaystyle \wp (S)}$; that is, ${\displaystyle {\mathcal {C}}\subseteq \wp (S).}$

A subset ${\displaystyle S}$ of a topological group ${\displaystyle X}$ is called a complete subset if it satisfies any of the following equivalent conditions:

1. Every Cauchy prefilter ${\displaystyle {\mathcal {C}}\subseteq \wp (S)}$ on ${\displaystyle S}$ converges to at least one point of ${\displaystyle S.}$
   • If ${\displaystyle X}$ is Hausdorff then every prefilter on ${\displaystyle S}$ will converge to at most one point of ${\displaystyle X.}$ But if ${\displaystyle X}$ is not Hausdorff then a prefilter may converge to multiple points in ${\displaystyle X.}$ The same is true for nets.
2. Every Cauchy net in ${\displaystyle S}$ converges to at least one point of ${\displaystyle S}$;
3. Every Cauchy filter ${\displaystyle {\mathcal {C}}}$ on ${\displaystyle S}$ converges to at least one point of ${\displaystyle S.}$
4. ${\displaystyle S}$ is a complete uniform space (under the point-set topology definition of "complete uniform space") when ${\displaystyle S}$ is endowed with the uniformity induced on it by the canonical uniformity of ${\displaystyle X}$;

A subset ${\displaystyle S}$ is called a sequentially complete subset if every Cauchy sequence in ${\displaystyle S}$ (or equivalently, every elementary Cauchy filter/prefilter on ${\displaystyle S}$) converges to at least one point of ${\displaystyle S.}$

• Importantly, convergence outside of ${\displaystyle S}$ is allowed: If ${\displaystyle X}$ is not Hausdorff and if every Cauchy prefilter on ${\displaystyle S}$ converges to some point of ${\displaystyle S,}$ then ${\displaystyle S}$ will be complete even if some or all Cauchy prefilters on ${\displaystyle S}$ also converge to points(s) in the complement ${\displaystyle X\setminus S.}$ In short, there is no requirement that these Cauchy prefilters on ${\displaystyle S}$ converge only to points in ${\displaystyle S.}$ The same can be said of the convergence of Cauchy nets in ${\displaystyle S.}$
  • As a consequence, if a commutative topological group ${\displaystyle X}$ is not Hausdorff, then every subset of the closure of ${\displaystyle \{0\},}$ say ${\displaystyle S\subseteq \operatorname {cl} \{0\},}$ is complete (since it is clearly compact and every compact set is necessarily complete). So in particular, if ${\displaystyle S\neq \varnothing }$ (for example, if ${\displaystyle S}$ a is singleton set such as ${\displaystyle S=\{0\}}$) then ${\displaystyle S}$ would be complete even though every Cauchy net in ${\displaystyle S}$ (and every Cauchy prefilter on ${\displaystyle S}$), converges to every point in ${\displaystyle \operatorname {cl} \{0\}}$ (include those points in ${\displaystyle \operatorname {cl} \{0\}}$ that are not in ${\displaystyle S}$).
  • This example also shows that complete subsets (indeed, even compact subsets) of a non-Hausdorff space may fail to be closed (for example, if ${\displaystyle \varnothing \neq S\subseteq \operatorname {cl} \{0\}}$ then ${\displaystyle S}$ is closed if and only if ${\displaystyle S=\operatorname {cl} \{0\}}$).

A commutative topological group ${\displaystyle X}$ is called a complete group if any of the following equivalent conditions hold:

1. ${\displaystyle X}$ is complete as a subset of itself.
2. Every Cauchy net in ${\displaystyle X}$ converges to at least one point of ${\displaystyle X.}$
3. There exists a neighborhood of ${\displaystyle 0}$ in ${\displaystyle X}$ that is also a complete subset of ${\displaystyle X.}$[25]
   • This implies that every locally compact commutative topological group is complete.
4. When endowed with its canonical uniformity, ${\displaystyle X}$ becomes is a complete uniform space.
   • In the general theory of uniform spaces, a uniform space is called a complete uniform space if each Cauchy filter in ${\displaystyle X}$ converges in ${\displaystyle (X,\tau )}$ to some point of ${\displaystyle X.}$

A topological group is called sequentially complete if it is a sequentially complete subset of itself.

Neighborhood basis: Suppose ${\displaystyle C}$ is a completion of a commutative topological group ${\displaystyle X}$ with ${\displaystyle X\subseteq C}$ and that ${\displaystyle {\mathcal {N}}}$ is a neighborhood base of the origin in ${\displaystyle X.}$ Then the family of sets

$${\displaystyle \left\{\operatorname {cl} _{C}N:N\in {\mathcal {N}}\right\}}$$

is a neighborhood basis at the origin in ${\displaystyle C.}$[23]

Uniform continuity

Let ${\displaystyle X}$ and ${\displaystyle Y}$ be topological groups, ${\displaystyle D\subseteq X,}$ and ${\displaystyle f:D\to Y}$ be a map. Then ${\displaystyle f:D\to Y}$ is uniformly continuous if for every neighborhood ${\displaystyle U}$ of the origin in ${\displaystyle X,}$ there exists a neighborhood ${\displaystyle V}$ of the origin in ${\displaystyle Y}$ such that for all ${\displaystyle x,y\in D,}$ if ${\displaystyle y-x\in U}$ then ${\displaystyle f(y)-f(x)\in V.}$